Page 226 - DSP Integrated Circuits

5.6 Round-Off Noise 211

Depending on how the arithmetic operations inside the butterfly are per-
formed, quantization errors may be avoided for the simple coefficient values ±1
and ±7. However, here we assume that we always get quantization error due to the
scaling coefficient. Hence, in practice, we can expect the round-off noise to be
somewhat lower than predicted by this simplified analysis.
It can be shown that the noise variance at an FFT output node is obtained by
adding the contribution from no more than NI such noise sources. Figure 5.19
shows which butterflies contribute noise to the output node X(0). Scaling the input
by a factor 1/2 does not introduce any noise since the input data word length is
shorter than the word length inside the FFT computations. Hence, the scaled
input values need not to be quantized.
For the unsealed FFT the magnitude of the frequency response from the out-
put of a butterfly to any output node of the FFT is unity, since I W\ =1. Hence, the

Figure 5.19 Butterflies that contribute to the noise at the first output node